Mon. Not. R. Astron. Soc. 424, 649–665 (2012) doi:10.1111/j.1365-2966.2012.21244.x

The complex case of V445 Lyr observed with Kepler: two Blazhko modulations, a non-radial mode, possible triple mode RR Lyrae pulsation, and more

, , E. Guggenberger,1 K. Kolenberg,2 3 J. M. Nemec,4 R. Smolec,1 5 J. M. Benko,˝ 6 C.-C. Ngeow,7 J. G. Cohen,8 B. Sesar,8 R. Szabo,´ 6 M. Catelan,9 P. Moskalik,5 K. Kinemuchi,10 S. E. Seader,11 J. C. Smith,11 P. Tenenbaum11 and H. Kjeldsen12 1Institut fur¨ Astronomie, Universitat¨ Wien, Turkenschanzstrasse¨ 17, A-1180 Vienna, Austria 2Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 3Instituut voor Sterrenkunde, Celestijnenlaan 200D, B-3001 Leuven, Belgium 4Department of Physics & Astronomy, Camosun College, Victoria, British Columbia V8P 5J2, Canada 5Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warszawa, Poland 6Konkoly Observatory, Research Center for Astronomy and Earth Sciences, PO Box 67, H-1525 Budapest, Hungary 7Graduate Institute of Astronomy, National Central University, Jhongli City, Taoyuan County 32001, Taiwan 8California Institute of Technology, Mail Stop 249-17, 1200 East California Boulevard, Pasadena, CA 91125, USA 9Departamento de Astronom´ıa y Astrof´ısica, Facultad de F´ısica, Pontificia Universidad Catolica´ de Chile, Av. Vicuna˜ Mackenna 4860, 782-0436 Macul, Santiago, Chile 10NASA Ames Research Center/Bay Area Environmental Research Institute, Mail Stop 244-30, Moffett Field, CA 94035, USA 11SETI Institute/NASA Ames Research Center, Moffett Field, CA 94035, USA 12Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark

Accepted 2012 May 4. Received 2012 April 30; in original form 2012 February 23

ABSTRACT Rapid and strong changes in the Blazhko modulation of RR Lyrae stars, as have recently been detected in high-precision satellite data, have become a crucial topic in finding an explanation of the long-standing mystery of the Blazhko effect. We present here an analysis of the most extreme case detected so far, the RRab star V445 Lyr (KIC 6186029) which was observed with the Kepler space mission. V445 Lyr shows very strong cycle-to-cycle changes in its Blazhko modulation, which are caused by both a secondary long-term modulation period and irregular variations. In addition to the complex Blazhko modulation, V445 Lyr also shows a rich spectrum of additional peaks in the frequency range between the fundamental pulsation and the first harmonic. Among those peaks, the second radial overtone could be identified, which, combined with a metallicity estimate of [Fe/H] =−2.0 dex from spectroscopy, allowed us to constrain the mass (0.55–0.65 M) and luminosity (40–50 L) of V445 Lyr through theoretical Petersen diagrams. A non-radial mode and possibly the first overtone are also excited. Furthermore, V445 Lyr shows signs of the period-doubling phenomenon and a long- term period change. A detailed Fourier analysis along with a study of the O − C variation of V445 Lyr is presented, and the origin of the additional peaks and possible causes of the changes in the Blazhko modulation are discussed. The results are then put into context with those of the only other star with a variable Blazhko effect for which a long enough set of high-precision continuous satellite data has been published so far, the CoRoT star 105288363. Key words: asteroseismology – methods: data analysis – techniques: photometric – stars: individual: KIC 6186029 (V445 Lyr) – stars: individual: CoRoT 105288363 – stars: variables: RR Lyrae.

1 INTRODUCTION RR Lyrae stars, which are low-mass helium-burning stars on the horizontal branch, were long thought to be rather simple radial pulsators. They follow a period–luminosity–colour relation which E-mail: [email protected] makes them valuable distance indicators, and because of their age

C 2012 The Authors Monthly Notices of the Royal Astronomical Society C 2012 RAS 650 E. Guggenberger et al. and evolutionary status, they are also used to study the formation a rich spectrum of additional modes in the region between 2 and and evolution of the Galaxy (Catelan 2009). They can oscillate in 4d−1. Both Fourier and O − C analyses are used to investigate the fundamental radial mode (type RRab), the first overtone (type the variability of the pulsation and the modulation (Section 4), RRc) or both modes simultaneously (type RRd), and their high and the results are compared to the case of CoRoT 105288363 amplitudes of up to 1.5 mag in V for RRab type stars made their in Section 6. Spectroscopy and theoretical Petersen diagrams are variability easy to discover, so they have been known since the end used to determine the fundamental parameters such as metallicity, of the 19th century. luminosity and mass (Section 5). Additionally, the new analytic Already more than a hundred years ago, however, it turned out that modulation approach for data analysis recently proposed by Benko,˝ there is an aspect to RR Lyrae stars which is not understood at all: Szabo&Papar´ o´ (2011) is applied to the data in Section 4.4. Blazhko (1907) found a ‘periodic change in the period’ of RW Dra, which he could not explain, and which still remains unexplained 2 BACKGROUND INFORMATION ON V445 LYR today. Shapley (1916) later found in his observations of RR Lyrae that the brightness of the maxima and the light-curve shape also V445 Lyr, with the coordinates RA 18h58m26s and Dec. 41◦3549 show periodic changes. With increasing data quality in the recent (J2000), is also known as KIC 6186029, or GR244, and has a Ke- past, the unsolved problem got even more severe, as it turned out pler magnitude of Kp = 17.4. Two publications from the pre-Kepler that not just a rather small fraction of exceptional RR Lyrae stars era exist for this target: Romano (1972) found it to be variable and were affected, but probably around 40–50 per cent of all RRab stars classified it as an RR Lyrae, and Kukarkin et al. (1973) included (Jurcsik et al. 2009; Benko˝ et al. 2010; Kolenberg et al. 2010). Also it into his name list of variable stars. Romano (1972) also lists the among RRc type stars, amplitude and phase modulation was found photographic brightnesses of maximum and minimum to be 15.3 to be surprisingly widespread (Arellano Ferro et al. 2012). and 17.3 mag, respectively, indicating a surprisingly large amplitude This so-called Blazhko effect was long thought to be a peri- of 2 mag, which is much higher than the amplitude observed even odic/regular phenomenon. Traditionally, only one Blazhko period during extreme Blazhko maxima in the modern data. This might at was assigned to each modulated star, and the phenomenon was ex- least partly be explained, however, by the difference between the pected to repeat in every Blazhko cycle, agreeing with the widely observed bandpasses. Unfortunately, no details of the observations used definition that ‘the Blazhko effect is a periodic amplitude and no light curves are given, and the forthcoming paper that was and/or phase modulation with a period of several tens to hundreds of announced by the author could not be found. We therefore cannot pulsation periods’. There were several reports about changes in the know if the observed amplitude of 2 mag is real or it is possibly due Blazhko modulation of various stars (see section 5 of Guggenberger to some observational errors. No error estimations of the observa- et al. 2011 for a recent summary), but those reports usually had to tions were given by Romano (1972). rely on sparse data with large gaps, so that it was impossible to say Since Kepler data have become available, two more publications when exactly a change took place and whether it happened continu- have dealt with V445 Lyr, both presenting the Kepler data up to ously or abruptly. The Blazhko effect was therefore still considered Q2: Szabo´ et al. (2010) listed V445 Lyr as a possible candidate for to be a strictly repetitive phenomenon with only some rare excep- the period-doubling phenomenon, and Benko˝ et al. (2010) already tions showing secondary modulation periods (e.g. CZ Lac; Sodor´ noted changes in the Blazhko modulation of V445 Lyr and reported et al. 2011) or changes on very long time-scales. It was not until the presence of radial overtones. the availability of ultraprecise data from space missions like CoRoT that strong and irregular cycle-to-cycle changes of a Blazhko star 3 KEPLER PHOTOMETRY were documented and that it became obvious that seemingly chaotic phenomena need to be accounted for when modelling the Blazhko The Kepler space mission was launched on 2009 March 6 into an effect. Earth-trailing heliocentric orbit (Koch et al. 2010). Its primary pur- While the detection of cycle-to-cycle changes in the Blazhko pose is the detection of Earth-sized planets in the habitable zone modulation posed a significant challenge for all classical models of solar-like stars through the transit method, which requires con- that required a clock-work-like behaviour, some new ideas were tinuous and ultraprecise photometry of over 150 000 stars for at published. Stothers (2006) suggested that transient small-scale mag- least 3.5 years. This is also the duration of the primary mission. netic fields modulate the turbulent convection inside the helium and Kepler therefore not only provides the longest continuous data sets hydrogen ionization zones, a mechanism which certainly could ex- ever observed for RR Lyrae stars, but also does so with the high- plain subsequent Blazhko cycles of different strengths. This sce- est photometric precision ever obtained, as a consequence greatly nario, however, was recently tested on the basis of hydrodynamical improving our knowledge about stellar pulsations. models by Smolec et al. (2011) who found that it was not possible The Kepler spacecraft carries a Schmidt telescope with an aper- to reproduce all observed properties of the light curve, even when ture of 0.95 m and 42 science CCDs which cover a field of view allowing a huge modulation of the mixing length. On the other hand, of about 115 deg2 (Jenkins et al. 2010). The photometric bandpass Buchler & Kollath´ (2011) successfully modelled both regular and ranges from 423 to 897 nm, thus avoiding the Ca II H&K lines in the irregular modulations by using the amplitude equation formalism. blue, and fringing due to internal reflection in the red. The Kepler In their models, a strange attractor in the dynamics causes chaotic band is therefore slightly broader than a combination of Johnson V behaviour. The 9:2 resonance between the fundamental mode and and R,andKepler magnitudes are usually about 0.1 mag from R for the ninth overtone that was found by Kollath,´ Molnar´ & Szabo´ most of the stars (Koch et al. 2010). (2011) to be the reason for the recently discovered phenomenon of Every quarter orbit, the spacecraft is rotated in order to keep the period doubling (Szabo´ et al. 2010) seems to play an important role solar panels oriented towards the Sun, and the radiator that cools in causing a Blazhko modulation. the focal plane towards deep space (Haas et al. 2010). Data from We present here a study of the most extreme case detected so different quarters are denominated Q1, Q2, etc. Each of the quarters far: V445 Lyr (KIC 6186029). This RRab star not only shows the which is used here has a time base of about 90 d, except Q1 which strongest cycle-to-cycle change found up to now, but also shows covers about 33 d.

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS The complex case of V445 Lyr 651

Each measurement is based on a 6.02 s exposure plus a 0.52 s occur due to image motion on the CCD or sensitivity changes. The readout time. To obtain long cadence (LC) data (29.4 min), 270 scaling and detrending which has to be performed before starting measurements are co-added; for short cadence (SC) data (1 min) the analysis is a delicate task especially for targets like RR Lyrae nine exposures are co-added (Christiansen et al. 2011). stars which have high amplitudes, long periods and changes in the The time of each measurement is given in truncated Barycentric amplitudes [see Celik et al. (2012) for a detailed discussion]. Here, Julian Date (HJD − 240 0000), and refers to the mid-point of the we removed linear trends which were determined from a running measurement. average separately for every quarter, and scaled every quarter to the same mean brightness. The continuity of the upper and lower envelope of the light curve was a good indicator of correct scaling. 3.1 The V445 Lyr data set Note that due to the gap during Q5, the continuity could not be Fig. 1 illustrates the data obtained for V445 Lyr. In this paper, we checked there. For this reason, the scaling of the data obtained after present LC data obtained in quarters 1–7, with a total time base of the gap might not exactly be consistent with the scaling applied 588 d. Because of the loss of module 3, which happened in 2010 before Q5. January, there are no data available for this target in Q5. Some The scaled and detrended data that were used in this analysis are smaller gaps are also present in the data as can be seen in Fig. 1. available as online material in the format displayed in Table 1 (see Some of them are due to unplanned safe mode events or loss of fine Supporting Information). point control, others are caused by the regular downlinks where the Fig. 1(a) shows the complete combined data set of V445 Lyr. spacecraft’s antenna is pointed towards Earth for data transmission, A total of 5.5 and 3.5 Blazhko cycles were observed before and and science data collection is interrupted. after the large gap, respectively, and 900 pulsation cycles were Kepler provides light curves in two different formats: raw flux observed. From panel (a) it is obvious that the Blazhko effect shows or corrected flux. The latter has been processed for planet transit strong and fast changes, with almost all observed cycles having a search by the Pre-search Data Conditioning (PDC) pipeline which is different appearance. Panel (b) compares in a phase diagram directly sometimes known to remove astrophysical features and which does a light curve from a Blazhko maximum with one obtained during not preserve all stellar variability. Hence here, and for the study of the subsequent Blazhko minimum, illustrating the extremely low variable stars in general, we make use only of the raw time series. pulsation amplitude that occurs during some, but not all, modulation As the spacecraft is rotated every quarter orbit, the target falls on minima. Note that due to the very small amplitude and the distorted different CCDs after each ‘roll’, resulting in differences in average light curve showing a double maximum (see also Fig. 1d) it would flux due to different sensitivity levels. Additionally, trends might not be possible to recognize the RR Lyrae nature of this star when

Figure 1. Panel (a) shows the complete light curve of V445 Lyr presented in this paper, including quarters Q1–Q7. Due to failure of a module, there are no data from Q5, resulting in the gap from MJD 55273 to 55372. Panel (b) illustrates the extreme Blazhko modulation by comparing a pulsation cycle at Blazhko maximum to one at Blazhko minimum. The light variation has an extremely low amplitude during this Blazhko minimum and shows a double maximum. In panel (c), a zoom into a region with period doubling is shown as an example and panel (d) emphasizes the distorted light-curve shape that occurs during some Blazhko minima.

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS 652 E. Guggenberger et al.

Table 1. The scaled and detrended data set of the thruster firings which are necessary to release the angular mo- V445 Lyr that was used for the analysis in this mentum that has built up in the reaction wheels, the spacecraft paper. Column 1 gives the truncated Barycentric momentarily loses the fine point control (Van Cleve & Caldwell Julian Date, column 2 gives the magnitude with 2009). This results in a missing data point during every desatura- the average shifted to zero and column 3 gives tion in the otherwise regularly spaced data, leading to a comb-like the quarter in which data were obtained. The full −1 table is available as Supporting Information with structure in the Fourier transform with a spacing of 0.335 d .In the online version of the paper. the spectral window, those peaks have a very low normalized am- plitude of only 0.004 mag. A zoom into the spectral window (see HJD − 240 0000 Magnitude Quarter lower panel of Fig. 2) reveals this comb of tiny peaks. When care- fully removing all high-amplitude peaks from the Fourier spectrum 54964.512 11 0.044 188 798 Q1 before turning to interpreting features with amplitudes which are 54964.532 54 0.038 912 686 Q1 orders of magnitude smaller as discussed in Section 4.1.5, those 54964.552 98 0.017 979 982 Q1 alias peaks are not expected to cause any trouble. − 54964.573 41 0.023 960 246 Q1 Other features in the spectral window function are two harmonics 54964.593 85 −0.099 222 117 Q1 of the monthly data downlink frequency, at 0.065 and 0.13 d−1, with ...... normalized amplitudes of 0.05 and 0.03 mag, respectively. Due to the time base of 588 d, the Rayleigh frequency resolution observing it only during Blazhko minimum where the peak-to-peak is 0.0017 d−1. amplitude can be as low as 70 mmag. Extremely low pulsation amplitudes during Blazhko minima were recently also reported by Sodor´ et al. (2012) for two of the RRab type stars, and the occurrence 4.1 Fourier analysis of the full data set of a strong bump during Blazhko minimum was noted in RZ Lyr As a first step, a Fourier analysis of the complete data set was (Jurcsik et al. 2012). In Fig. 1(c), a small part of the light curve performed, keeping in mind that changes in the fundamental pe- around JD 245 5386 is shown, illustrating the phenomenon of period riod might occur during the time of the observations and that the doubling (alternating higher and lower light maxima) which was changes of the Blazhko modulation cause a large number of peaks recently discovered in Kepler Blazhko stars (Szabo´ et al. 2010) and close to the classical pulsation and modulation components. The which is also present in V445 Lyr. Fourier analysis of the full data set, however, is necessary to ob- tain an overall picture of the pulsation and modulation properties of 4 ANALYSIS AND RESULTS V445 Lyr to find the mean values of the pulsation and modulation periods and to detect possible long-term periodicities which can be Due to the quasi-continuous coverage and the high photomet- resolved with a long time base only. ric precision, the conditions for a Fourier analysis are more than The Fourier analysis of the complete data set was performed favourable. The spectral window function (see Fig. 2) is almost per- with PERIOD04 (Lenz & Breger 2005) and then checked with fect, without any alias peaks visible at first glance. Because of the SIGSPEC (Reegen 2007). The results agreed within the errors. The Earth-trailing orbit, no orbital frequencies like the ones that cause Fourier analysis revealed a mean pulsation period of 0.513 075 ± trouble in the data from many other space missions are present here. 0.000 005 d and a mean Blazhko period of 53.1 ± 1 d with the A well-known feature in the Kepler data is the momentum desat- ephemeris uration of the reaction wheel, which happens every 2.98 d. During HJD(Tmax,pulsation) = 245 5550.514 + 0.513 075Epulsation,

HJD(Tmax,Blazhko) = 245 5534.2 + 53.1EBlazhko. Note that the presence of close peaks has a great influence on the fitting and frequency optimization procedure, and the errors are therefore larger than would normally be expected for a Fourier fit to data of the given quality.

4.1.1 Multiplet components The Blazhko multiplets (i.e. the pattern of peaks which is typical for Blazhko stars with peaks at the positions kf0 ± nf B, with k and n being integers denoting the harmonic order and the multiplet order, respectively, and with f 0 and fB denoting the fundamental and the Blazhko frequency) were found to be very asymmetric in amplitude. Much higher amplitudes appeared on the right-hand (higher frequency) side than on the left. This is the more common case, which is observed in three-fourths of all Blazhko RRab stars (Alcock et al. 2003). Components were detected up to quintuplet order on the right-hand side, while on the left-hand side of the main Figure 2. Spectral window function of the complete V445 Lyr data set. pulsation component only one side peak (i.e. a triplet component, Lower panel: magnification of the region between 5 and 7 d−1, showing n = 1) could be found. In some orders, peaks were found near the some examples of the low-amplitude features caused by the reaction wheel. positions that would be expected for septuplets, but the deviations

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS The complex case of V445 Lyr 653 from the exact frequency values were considered too big to identify the peaks safely as septuplet components. In addition to the classical Blazhko multiplets around the fun- damental mode and its harmonics, a secondary modulation com- ponent, fS, could also be identified. It manifests itself as a series of additional peaks next to the classical modulation side peaks of the Fourier spectrum, appearing in every order with a spacing of 0.006 98 ± 0.000 27 d−1, indicating a secondary modulation period of 143.3 ± 5.8 d. The secondary peaks appearing on the right-hand side of the classical peaks show surprisingly high amplitudes, while the secondary peaks to the left of the classical multiplet have small amplitudes and could only be detected after pre-whitening a large number of higher amplitude peaks. Interestingly, the peaks belong- ing to this additional series could also be detected at higher multiplet orders (n = 3) than the classical multiplet, making it difficult to ex- plain the peaks as combination frequencies. Their amplitudes seem to decrease less quickly with increasing harmonic order k (see also Section 4.1.3), making them easier to spot at high orders. Fig. 3 shows the Fourier transform of the data, providing also zooms into the regions around the third, fourth and fifth harmonic order where both the classical multiplet and the additional com- ponents can be seen. The highest peaks among the multiplet com- ponents, which can be seen even before pre-whitening the original data, are marked with arrows. Fig. 4 illustrates the pattern of detected frequencies in the vicin- ity of the fundamental pulsation and its harmonics in the style of an ‘echelle’ diagram. In this diagram, which is similar to the di- agrams used to unveil equally spaced peaks in helioseismology, the frequency of each peak is plotted against (f modulo f 0), i.e. f/f0 − INT(f/f0)orf/f0 − INT(f/f0) − 1 for peaks to the left of the harmonic, therefore clearly revealing patterns which repeat in every harmonic order. Peaks belonging to the same group of combinations align in vertical ridges. We stress that unlike in the helioseismic application, where the ridges denote different radial or- ders of same degree, in this case the echelle diagram only serves the purpose of displaying in a very practical and easy way the repeating patterns in different harmonic orders of non-sinusoidal fundamental radial pulsation.

4.1.2 Deviation of the harmonics Due to the non-sinusoidal light-curve shape typical for RR Lyrae stars, harmonics of the fundamental mode are expected to appear at frequencies kf 0,wherek is an integer denoting the harmonic order. The classical Blazhko multiplets in the modulated stars are spaced equidistantly, implying frequency values of kf0 ± nf B with n denoting the multiplet order. Long-term period changes and close peaks caused by irregular phenomena, however, can distort this fre- quency pattern. When analysing time series of Blazhko RR Lyrae stars, there are two options for fitting the data: one is to fix the fre- quencies of the harmonics and Blazhko multiplets to their expected values of kf0 +nf B, thereby reducing the number of free parameters in the fit. The other option is to let all parameters, including the fre- quency values, free. When the latter option was applied to this data Figure 3. Fourier transform of the complete V445 Lyr data set (upper set, the harmonics were observed to deviate systematically and sig- panel). In the lower panels (b, c and d), the regions around the harmonics kf with k = 3, 4 and 5 are shown in detail. Black arrows indicate the peaks nificantly from their expected values, which can also be noticed as 0 belonging to the classical Blazhko multiplet with kf − f , kf , kf + f a slight rightwards tilt of the ridges in the echelle diagrams (Fig. 4). 0 B 0 0 B and kf0 + 2fB, while grey arrows indicate some of the peaks belonging Normally, one would expect that this is simply caused by a wrong to the secondary multiplet which within itself has a spacing equal to the value of f 0, but in this case, no value of f 0 could be found which Blazhko frequency, but is shifted with respect to the classical multiplet by could solve the issue, i.e. every detected harmonic, when divided by 0.006 98 d−1. Note that from order k = 4 onwards, the highest amplitude its order, required a different f 0. We therefore decided in favour of peak in this order is a peak belonging to the secondary multiplet.

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS 654 E. Guggenberger et al.

Figure 5. Amplitudes of all components versus harmonic order. Compo- Figure 4. ‘Echelle’ diagram of the peaks detected in the vicinity of the nents belonging to the classical Blazhko multiplet are shown with green fundamental pulsation mode and its harmonics. Frequency is plotted versus symbols, peaks which are combinations with fS are shown in orange, and f modulo f , so that regularly spaced peaks which appear in every order 0 components with a 2fS term are shown in magenta. It can be seen that the align in vertical patterns, making it easy to identify combinations with f 0. amplitude of the f multiplets decrease less steeply than those of the classi- kf ±nf S Peaks belonging to the category 0 B are shown as black squares, peaks cal Blazhko multiplet, reaching the same amplitude in the fourth order and kf ± nf + f of the group 0 B S are shown as grey circles and components of dominating the frequency spectrum for higher orders. kf0 + nf B + 2fS are plotted with light-grey triangles. Open circles denote unidentified and/or unresolved peaks, some of which originate from the tions with the term 2f could be detected, and their amplitudes are non-repetitive nature of the modulation and long-period changes. S plotted in Fig. 5. They also show a very slow amplitude decrease. the more pure approach and did not fix the frequency values to the expected positions, but left all parameters free in the fit of 4.1.4 Number of relevant frequencies the complete data set. The problem disappeared, however, when analysing different subsets of data separately (see Section 4.2), and Due to the dense spectrum of peaks which is caused by the cycle- we therefore suspect it to be either the result of period changes which to-cycle changes of the Blazhko effect, an analysis in the classical take place during certain seasons (see also Section 4.3) and/or close sense, i.e. taking into account all frequencies down to a certain unresolved peaks which are known to strongly influence the results signal-to-noise ratio level or a certain significance criterion, might of both the Fourier analysis and the multisine fitting procedure. not be optimal in a case like V445 Lyrae, as it does not yield mean- ingful results. A large number of the detected peaks is likely to be the result of ‘stellar noise’ caused by irregular and/or long-periodic phenomena, and many of them are not resolved with the available 4.1.3 Amplitudes versus harmonic order time span. Tests revealed that as many as 771 frequencies can be It is a well-known fact that in Blazhko RRab stars the amplitudes of found when performing an analysis until the generally adopted crite- the multiplet side peaks decrease less rapidly with harmonic order ria of significance are reached. Many of them were not resolved, and than that of the main component. This was first described by Jurcsik many could not be attributed to any combination of other modes, and et al. (2005), and then confirmed for other well-studied stars such did not show repeating patterns in the echelle diagrams. Therefore, as SS For (Kolenberg et al. 2009), RR Lyr (Kolenberg et al. 2011) instead of choosing the classical approach, the analysis was stopped and CoRoT 105288363 (Guggenberger et al. 2011). As expected, after a certain number of the highest peaks in every harmonic order the result is the same for V445 Lyr (see Fig. 5), but in an extreme were found and pre-whitened, usually around 20 peaks per order. It way with the amplitude of the right-hand side peak exceeding that turned out that after subtracting approximately 20 peaks in a given of the main component as early as in the third order. Additionally, harmonic order, no meaningful combinations could be identified the amplitudes of the secondary modulation multiplet, i.e. combi- among the following peaks, and many unresolved peaks appeared. nations with the secondary modulation fS, could be studied in this In Fig. 4 only the highest peaks of every order are shown, already case. It turned out that the amplitudes of the secondary multiplet including some unresolved peaks which could not be avoided due components decrease even less steeply, therefore dominating the to their high amplitudes. As the Nyquist frequency of LC data is Fourier spectrum from the fourth order onwards. The strongest sig- 24.4 d−1, 12 harmonic orders could be observed, and 239 frequen- nal then comes from the combination f0 + fB + fS, i.e. the peak on cies were subtracted around the main pulsation components until the right-hand side of the right triplet component. Also, combina- the attention was turned towards the additional peaks which are

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS The complex case of V445 Lyr 655 present in the region between the harmonics of the fundamental mode (see the next section). Altogether, 239 frequencies were included, of which one is the fundamental mode, nine are harmonics of the fundamental mode, 104 are combinations of f 0 or its harmonics with the Blazhko frequency fB and/or the secondary modulation fS and 125 are unidentified.

4.1.5 Additional frequencies: radial overtone and non-radial pulsation It was already noted by Benko˝ et al. (2010) that V445 Lyr shows a rich spectrum of frequencies in the region between the harmonics of the fundamental mode. Those frequencies are not at all typical for ab-type RR Lyrae stars and have never been detected in such a large number in an RR Lyrae star. Some peaks were suspected to be radial overtones by Benko˝ et al. (2010), and also frequencies which Figure 7. Echelle diagram of the additional frequencies, showing how the are half-integer multiples of the fundamental mode are expected patterns repeat in every harmonic order. Shaded boxes indicate the ranges to appear in this region as a consequence of the period-doubling which would be typical for the first and second overtones (O1 and O2, phenomenon as described by Szabo´ et al. (2010). However, there is respectively), as well as for the HIFs. more than this in the case of V445 Lyr. Fig. 6 shows the frequency spectrum after subtraction of the In a Fourier analysis of the relevant frequency regions, 80 peaks relevant peaks around the multiples of the fundamental pulsation (including combinations) were considered significant and were sub- as discussed in the previous section (their former positions are tracted. A closer inspection of the result revealed that the dominant marked with arrows). The additional frequencies can clearly be peaks formed combinations not only with the fundamental mode seen to be the dominant signal with an amplitude of 3.7 mmag for but also with the Blazhko multiplet peaks (including quintuplets!) the highest peak. The region between the fundamental mode and and in some cases also with the peaks belonging to the secondary its first harmonic is indicated with a grey box and enlarged in the multiplet. Negative combinations such as fN − f0 − fB also occur. insert (panel b). Four frequency regions with enhanced signal can be Significant combination peaks can be traced up to the fifth harmonic noted in the enlargement: around 2.65, 2.8, 2.9 and 3.33 d−1.This order, but an excess in signal is visible in the Fourier spectrum even pattern repeats in every harmonic order, indicating combinations at much higher orders (see Fig. 6). In Fig. 7, an echelle diagram is of the frequencies with the fundamental mode and its surrounding plotted for the additional peaks, clearly showing the combinations peaks. The presence of combinations is a strong evidence that the with the fundamental mode aligned in vertical patterns. Shaded additional signal is not introduced by a possible background star. boxes indicate the typical regions in which overtone modes and half-integer combination frequencies would be expected to be sit- uated. Please note that it was shown by Szabo´ et al. (2010) that due to the onset and offset of the period-doubling phenomenon, the half-integer frequency (HIF) peaks are not necessarily located at the exact positions of the half-integer multiples, but might deviate by several per cent. Therefore, the shaded box at (f modulo f 0) = 0.5 in Fig. 7 has a distinct width. In V445 Lyr, the frequency at 2.9256 d−1 deviates by only −1 0.0021 d (i.e. 0.07 per cent) from the exact value of 3f 0/2. Given the fact that clear signs of period doubling are indeed visible in the light curve (see Fig. 1c), and considering the above-mentioned find- ings of Szabo´ et al. (2010), it is quite safe to interpret this frequency as an HIF caused by period doubling. We hereafter refer to it as fH. Four significant HIFs (which can also be interpreted as combi- nations of fH with f 0) were found in the Fourier spectrum: 3f 0/2, 5f 0/2, 9f 0/2 and 11f 0/2. Combinations with the Blazhko frequency, both positive and negative, could also be identified (see Table 2 for a complete list). Another interesting feature is the peak at 3.3307 d−1 which shows Figure 6. Fourier spectrum after pre-whitening 239 frequencies in the vicin- a frequency ratio of f 0/f 2 = 0.585 with the fundamental mode, ity of the fundamental mode and its harmonics. Very clearly, additional fre- and which we hereafter refer to as f . Its period ratio is typical quencies can be seen between the harmonic orders and their combinations 2 for the second overtone. Peaks with similar period ratios have al- with f 0 which are present up to high harmonic orders. Arrows mark the places where the harmonics and the Blazhko as well as the secondary multi- ready been reported for several RR Lyrae stars. Poretti et al. (2010) plets were located before pre-whitening. Some signal remains around their were the first to find them in the star CoRoT 101128793. The fre- positions, as discussed in Section 4.1.4. The insert (panel b) shows a zoom quency ratio in their study was 0.584 with the fundamental mode −1 into the region between the fundamental mode and the first harmonic (2– at f 0 = 2.11895 d . They interpreted the peak as the second ra- 4d−1), indicated as a grey square. dial overtone. The same authors also re-analysed the data of V1127

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Table 2. List of the highest peaks in every one of tone (O2) excited. Guggenberger et al. (2011) found evidence for the four regions of excess signal, and their combi- the second overtone in CoRoT 105288363, a Blazhko star with nations. rapid changes in the Blazhko effect, and Nemec et al. (2011) found the second overtone in KIC 7021124, therefore providing another Frequency Amplitude −1 example of a non-modulated RRab star pulsating in both F and O2 (d ) (mmag) −1 (f 0 = 1.606 445d , f 0/f 2 = 0.593). fN 2.7719 3.67 It is interesting to note how different the stars are for which the fN + f0 4.7211 1.58 second overtone has been documented so far: their fundamental −1 fN + 2f0 6.6778 0.74 frequencies range from about 1.6 to 2.8 d , covering almost the f + f N 3 0 8.6196 0.51 full bandwidth of RRab pulsation, and with respect to stability f − f N 0 0.8228 0.75 they range from non-modulated stars with almost perfectly regular

fN + fB 2.7895 1.64 RRab pulsation (V350 Lyr, KIC 7021124) to Blazhko stars with a fN + f0 + fB 4.7389 1.43 rather regular Blazhko effect (CoRoT 101128793 and V1127 Aql) fN + 2f0 + fB 6.6880 1.22 and finally to modulated stars that show dramatic changes of their f + f + f N 3 0 B 8.6372 0.83 Blazhko modulation (CoRoT 105288363 and V445 Lyr). Moreover, f + f + f N 4 0 B 10.5864 0.54 they cover a significant range of Blazhko periods, from 16.6 d to f + f + f N 5 0 B 12.5359 0.33 more than 200 d, as estimated for V2178 Cyg. fN − f0 − fB 0.8044 0.69 The combinations of f 2 in V445 Lyr deserve some special atten- fN + 2fB 2.8091 2.24 tion. While as many as 32 peaks are detected near the positions of f + f + f N 2 0 2 B 6.7067 0.87 f 2 + kf 0, and clearly an excess of signal is visible in every harmonic f + f + f N 3 0 2 B 8.6559 0.57 order (see Figs 6 and 7), it was not possible to identify most of the f + f + f f N 5 0 2 B 12.5543 0.34 detected peaks as exact combinations with the known frequencies. fN + 2f0 + fS 6.6778 0.74 For fH (which was discussed in the previous paragraph), 12 out of f + f + f + f N 2 0 B S 6.6964 0.76 16 peaks could be attributed to combinations while for f 2, only three f + f + f + f N 3 0 B S 8.6456 0.81 combinations with f 0 were found at their exact positions. All the f + f + f + f N 3 0 3 B S 8.6828 0.53 other peaks in the vicinity of the combinations deviated too much from the calculated values to be safely matched with the combi- f 1 2.6676 2.80 nations. This is especially remarkable as the amplitudes of those f 1 + f 0 4.6166 1.08 peaks are surprisingly large in higher harmonic orders compared to f + f + f 1 0 B 4.6362 1.16 the combinations of the other additional frequencies. From Fig. 6 it f + 2f + f 6.5851 1.02 1 0 B is obvious that in the first harmonic order, f has a small amplitude f + 3f + f 8.5343 0.77 2 1 0 B compared to the other additional frequencies, while at the orders f1 + 4f0 + fB 10.4838 0.45 4–5 they become equal, and at higher orders, the peaks in the area f1 − f0 + fB 0.7374 1.16 around f 2 + kf 0 are the dominant features. This is also illustrated fH 2.9256 2.55 in Fig. 8, where the amplitudes of the safely identified peaks ver- f + f H 0 4.8736 0.87 sus frequency are shown. The large number of unidentified peaks f + f H 3 0 8.7743 0.60 around f might indicate that the amplitude of f is variable, either f + f 2 2 H 4 0 10.7234 0.43 irregularly or on a time-scale other than the Blazhko frequency. This fH − fB 2.9070 2.29 will be discussed in more detail in Section 4.2.1. fH + f0 − fB 4.8607 0.89 The highest amplitude peak among the additional frequencies f + f − f − H 2 0 B 6.8060 0.95 (3.7 mmag) is the one at 2.7719 d 1. This peak was interpreted f + f − f H 3 0 B 8.7549 0.57 as the first overtone by Benko˝ et al. (2010), but its ratio with the fH + 4f0 − fB 10.7029 0.40 fH − f0 − fB 0.9579 1.01 fH + f0 + fB 4.8924 1.00 fH + 2f0 + fB 6.8413 0.71 fH + 3f0 + fB 8.7905 0.58

f 2 3.3307 1.43 f 2 + f 0 5.2781 0.85 f 2 + 2f 0 7.2268 0.79 f 2 + 3f 0 9.1760 1.04

Aql (Chadid et al. 2010) and MW Lyr (Jurcsik et al. 2008) and −1 found frequency ratios of 0.582 (f 0 = 2.8090d ) and 0.588 (f 0 = 2.5146d−1), respectively. In the sample of Kepler RRab stars, Benko˝ et al. (2010) reported the presence of the second overtone in four −1 different stars: V354 Lyr (a Blazhko star with f 0 = 1.780 37d ), = V2178 Cyg (a Blazhko star with f 0 2.054 23), V445 Lyr (the sub- Figure 8. Amplitudes of the additional frequencies and their combinations = ject of this paper) and the non-modulated RRab star V350 Lyr (f 0 with f 0 versus frequency. While the amplitudes of fN and f 1 decrease 1.682 82) which was the first example of a non-Blazhko double- exponentially with harmonic order, the amplitude of f 2 remains almost mode RR Lyrae star with the fundamental (F) and the second over- stable. Amplitude errors are smaller than the symbols.

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS The complex case of V445 Lyr 657 fundamental (f0/fN = 0.703) is very low compared to the canonical value of 0.74–0.75. Note that the OGLE III RR Lyrae stars in the Galactic bulge have f 0/f 1 values going down to about 0.726 only (see Soszynski et al. 2011). Extremely high metallicity values would be necessary according to models (Popielski, Dziembowski & Cassisi 2000; Szabo,´ Kollath´ & Buchler 2004; Smolec & Moskalik 2008) to fit this frequency ratio with the first overtone. New spectroscopic re- sults revealed, however, that the metallicity of V445 Lyr is [Fe/H] = −2.0 ± 0.3 (see Section 5), rendering it impossible to explain this frequency with a radial overtone mode. fN is therefore most likely to be a non-radial mode. We note that Van Hoolst, Dziembowski & Kawaler (1998) found in their non-adiabatic and non-radial cal- culations the excitation of non-radial modes in the vicinity of the radial mode in RR Lyrae stars to be very likely, and Dziembowski & Cassisi (1999) noted in their model survey the presence of strongly trapped non-radial modes with very high growth rates near the first overtone. −1 f Figure 10. Petersen diagram for the peak at 2.6676 d illustrating that a Among the additional modes in V445 Lyr, N is the one that metallicity of Z = 0.004 would be necessary to identify this peak as the first shows the clearest pattern of combination frequencies: of 20 peaks radial overtone mode. Models around the spectroscopically derived value which were found significant in relation to fN, all 20 could be un- of Z = 0.0003 (see Section 5) are also shown for comparison. Different ambiguously identified as exact combinations with f 0, the Blazhko symbols indicate different luminosities and different line styles indicate frequency and the secondary modulation frequency (see also different masses (see also Section 5). Models were calculated with the Table 2 and Fig. 9, which illustrate the regular pattern of com- Warsaw codes (Smolec & Moskalik 2008). bination peaks). We note that the possible non-radial mode, which was found by Chadid et al. (2010) in V1127 Aql, has a very similar frequency ratio (0.696), may be hinting at a possible systematic spectroscopic value (Z = 0.0002, see also Section 5). Moreover, − − = preference in non-radial mode excitation in RR Lyrae stars. the average frequency values are quite far (2f 1 f 0 f 2 0.055) There remains, however, the fourth region of increased signal from the resonance condition that could explain the presence of the with a main peak at 2.6676 d−1, which, with a frequency ratio of second overtone by resonant excitation, and which would also have f 0/f 1 = 0.731 is in principle in the possible range of the first overtone the power to shift the frequency away from the expected value in pulsation. One has to note that double-mode RR Lyrae stars usually the Petersen diagram. We note, however, that the frequency values follow a well-defined empirical sequence in the Petersen diagram; in of f 1 and f 2 are not strictly constant during the observed time span, other words, there is a relation between f 0 and f 0/f 1 (see Popielski but undergo irregular fluctuations. We performed a time-dependent et al. 2000; Soszynski et al. 2009). If the peak at 2.6676 d−1 is Fourier analysis (see also Section 4.2.1) and found the resonance indeed the first overtone, V445 Lyr would be an exception to this criterion to be fulfilled occasionally. For f 1, one combination with f 0 f relation, which is very unlikely. On the other hand, outliers from and five combination frequencies with f 0 and B were found, leaving the sequence similar to V445 Lyr have recently been reported by the other six significant peaks unidentified. One of them, a peak at − Soszynski et al. (2011) in the OGLE III survey of the Galactic bulge. 2.639 d 1 (with a frequency ratio of 0.739 with the fundamental The metallicity needed to reproduce a frequency ratio of f 0/f 1 = mode), would fulfill the resonance criterion, but its amplitude is 0.730 with models (Z = 0.004, see Fig. 10) is much larger than the only 1.9 mmag, compared to 2.8 mmag of f 1. We therefore conclude that the identities of the peaks at 2.6676 and 2.639 d−1 cannot be unambiguously assessed. The highest peaks in every region and their combinations are listed in Table 2. Altogether, 80 frequencies were found to be sig- nificant in the vicinity of the additional peaks: four of these were independent frequencies, 40 were combinations of these indepen- dent terms with f 0, fB and/or fS, and 36 were peaks which could not be identified as combinations. Together with the 239 frequencies found in the vicinity of the fundamental mode and its harmonics (see Section 4.1.4), this results in a total of 319 frequencies included in the analysis.

4.2 Separate analysis of subsets To study the time-dependent behaviour of the frequency pattern, we used both PERIOD04 and the time-resolved feature of SIGSPEC (Reegen 2007), which makes it possible to analyse large numbers of data Figure 9. Details of the echelle diagram, showing only the vicinity of the subsets in an automated way. The top panel of Fig. 11 illustrates the mode f N. All detected peaks could clearly be identified as combinations of variation of f with time as calculated with SIGSPEC for overlapping f N with either the classical Blazhko multiplet or the secondary multiplet. The 0 fact that no unidentifiable peaks are among the highest ones points towards subsets of 15 d. With PERIOD04, we analysed larger subsets of data, a very stable amplitude of this mode, compared to the other additional peaks consisting of one or more quarters each. During this process we note that were found in this star. that in the subset containing Q1 to Q4 (Blazhko period 53.9 d), no

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rule that can predict a high f 2 amplitude. This irregular amplitude variability can explain the numerous peaks around f 2 and its combi- nations which can be seen in the echelle diagram (Fig. 7) and which are discussed in Section 4.1.5. We also check the temporal variations of the frequency values of all the additional frequencies and find, in addition to the variation of f 0 which is plotted in Fig. 11(a), slight irregular variations of f 1 and f 2, which lead to an exact parametric resonance during some time intervals in the observed data. This might result in the momentary and transient excitation of f 2 which is seen in Fig. 11. We note that from the theoretical point of view, a resonance is necessary to excite the second overtone in this parameter range, because otherwise this overtone would not become unstable.

4.2.2 Variation of the Blazhko modulation parameters The modulation parameters are normally used to describe the prop- erties of the Blazhko modulation of a given star. The traditional parameters are Rk = A+/A− where A+ and A− are the amplitudes of the peak on the higher frequency and the lower frequency sides of the triplet, respectively, and the phase difference ϕk = ϕ+ − ϕ−. The parameter k denotes the harmonic order. Moreover, the asym- metry parameter Q = (A+ − A−)/(A+ + A−), which was introduced by Alcock et al. (2003), is widely used, as is the power difference 2 2 2 A = A+ − A−. This parameter was recently shown by Szeidl & Jurcsik (2009) to be the physically most meaningful one, as it Figure 11. Variation of f 0 over the complete data set (panel a). Middle panels show the variation of the amplitude (panel b) and the significance is directly correlated to the phase difference between the amplitude (panel c) of the second overtone f 2, calculated for bins of 15 d duration and the phase modulation components in their model of modulated with a step width of 2 d. The bottom panel (d) shows the light curve for oscillation. orientation. The times of significant f 2 amplitude are marked with shaded As the main aspect of V445 Lyr is the variability of the Blazhko boxes. cycles, we show here not only the average Fourier parameters for ev- ery order (which are listed in Table 3) but also their time-dependent behaviour in Fig. 12. To calculate the modulation parameters for deviation of the harmonics as described in Section 4.1.2 is observed. every cycle, the data set was divided into bins, starting and ending This phenomenon seems to occur only in quarters Q6 and Q7 where around Blazhko minimum. The length of the bins was about 60 d, the Blazhko periods found in separate analyses of the quarters were with the exception of the last bin which was only 50 d. Some over- 79.8 and 80.4 d, respectively. lap was allowed to guarantee frequency resolution of the Blazhko multiplet. It was an important result for the star CoRoT 105288363 that the 4.2.1 Stability of the additional frequencies phasing between the two types of modulation was found to change (see section 5.3 of Guggenberger et al. 2011). Such a phase change The results discussed in Section 4.1.5 hint towards a variability of is also indicated by the variation of A2 of V445 Lyr (bottom panel the amplitude of f 2, and we therefore studied the temporal evolution of Fig. 12). of this peak in detail. Using the time-resolved mode of SIGSPEC,we performed a Fourier analysis of overlapping bins with a duration of 15 d in steps of 2 d, limiting the frequency range to the region around 4.2.3 Fourier parameters f 2. The resulting amplitudes are plotted in Fig. 11(b). While there ϕ is a clear variation of the amplitude ranging from 2 to ∼7 mmag, As the so-called Fourier parameters Rk1 (amplitude ratio) and k1 no clear periodicity is discernible. We also performed a Fourier (epoch-independent phase difference) are considered useful tools to analysis on the resulting amplitude curve and found no significant frequency of variability. We note that in Fig. 11, all values of A(f 2) were included, regardless of the significance of the frequency. Only Table 3. Overall modulation parameters of V445 Lyr found from the fit to the complete bins with an insufficient number of data points and/or bad frequency data set for the first six harmonic orders. resolution were discarded. Therefore we provide as a supporting plot 2 the time-dependent significance of the peaks in panel (c), where the kRk ϕk Qk Ak most commonly used significance criterion of sig ≥ 5 is indicated with a dashed line. 1 2.112 −0.87 0.357 0.005 779 For better orientation, the bottom panel shows the full data set 2 3.179 0.27 0.521 0.001 900 (light curve) of V445 Lyr, and shaded boxes indicate the regions of 3 3.991 −0.24 0.599 0.000 445 4 3.806 −0.08 0.584 0.000 114 enhanced f 2 amplitude. There seems to be a preference for phases 5 5.382 −0.30 0.687 0.000 037 close to Blazhko minima, which are also the phases where the min- 6 4.715 −0.39 0.650 0.000 011 ima of the fundamental frequency f 0 occur, but there is no strict

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are defined as Rk1 = Ak/A1 and ϕk1 = ϕk − kϕ1. The data were subdi- vided into 239 bins of 2 d duration, therefore containing about four pulsation periods. On such short time-scales, the Blazhko effect is expected to play only a minor role. The effect of period doubling, however, causes a large error in some bins which are more affected by this effect than others. A fit including the fundamental pulsation and 10 harmonics is calculated for each bin, and the results are displayed in Fig. 13 (amplitude ratios are shown in the left-hand panels and phase differences in the right-hand panels). The average values of the parameters (derived from the complete data set) are given in Table 4. Some interesting details are immediately obvious: while the am- plitude A1 of 0.18 mag is quite small compared to other RR Lyrae stars, but still in the normal range, the amplitude ratios Rk1 are significantly smaller than those of the non-modulated stars (for Figure 12. Modulation parameters of V445 Lyr versus time. The top panel comparison, see fig. 6 in Nemec et al. 2011). The sharp upward shows the variation of R1, which is defined as the amplitude ratio between spikes are an intriguing feature in the R21 variation, which occur the right and the left modulation side peak, and the bottom panel displays only during some of the observed Blazhko minima. When looking A2 2 the variation of the power difference 1. The variation of A1 points at the phases one notes that, while ϕ1 has a smooth periodic vari- towards a shift in the phasing between the two types of modulation. ation, the phases ϕ2, ϕ3 and ϕ4 show a more or less continuous progression (with exceptions in some cycles), leading to apparent study and compare the properties of RR Lyrae stars, we calculated phase jumps in the ϕk1 parameters. their time variability for V445 Lyr. In contrast to the modulation The stability of the results was tested by also using other bin- parameters discussed in the previous section, they describe the pul- nings, and the size of the bins was found to play only a minor sation rather than the modulation properties. The Fourier parameters role.

Figure 13. Variation of the Fourier parameters during the observed time span, calculated for subsets of 2 d duration each. Left-hand panels: at the top, the amplitude A1 is displayed to indicate where Blazhko maxima and minima are located, while lower panels show the amplitude ratios R21, R31, R41 and R51. Right-hand panels: at the top, the phases ϕ1, ϕ2, ϕ3 and ϕ4 are plotted, while the lower panels show the phase differences ϕ21, ϕ31 and ϕ41 in the sine frame. Where error bars are not visible, they are smaller than the symbols.

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Table 4. Average Fourier parameters of V445 Lyr.

Parameter Value Parameter Value

A1 0.184 ϕ1 5.84 R21 0.268 ϕ21 2.34 R31 0.096 ϕ31 4.95 R41 0.033 ϕ41 1.08 R51 0.013 ϕ51 3.69

Figure 15. O − C diagram of the complete data set of V445 Lyr (top panel), and residuals after subtracting a fit with the Blazhko frequency, its harmonics and the secondary modulation frequency (bottom panel). To guide the eye, a line through the means of the residuals in bins of 10 maxima has been plotted.

ing double maxima (the double maxima are illustrated in Figs 1b and d). The determination of the time of the light maximum be- comes ambiguous here, depending on the maximum that is chosen. The original maximum seems to move ‘to the left’ (causing negative ϕ Figure 14. 1 versus A1 diagram of V445 Lyr. In this plot it can be seen O − C values) while a bump on the descending branch gets stronger how various observed cycles differ from each other. Both the phase modula- and takes the role of the maximum for the next Blazhko cycle. tion component and the amplitude modulation component change over the This also explains the rather abrupt transition from very low to observed time span. Moreover, their relative phasing changes from cycle to high O − Cvalues. cycle. We also performed a Fourier analysis on the O − Cdatato 4.2.4 Loop diagrams check whether the secondary modulation is also present in the phase variation. We clearly found the Blazhko frequency fB,in −1 A good indicator for the relative contributions of phase modulation this case 0.0184 d , as well as the harmonics 2fB and 3fB in the and amplitude modulation, and for the phasing between those two O − C curve, which are introduced by the non-sinusoidal variation types of modulation, is the A1 versus ϕ1 diagram. The resulting of the O − C curve. The secondary modulation, with a value of −1 loops for V445 Lyr are plotted in Fig. 14. The direction of motion is fS = 0.0064 d , as well as the combination peak fB + fS,was indicated with arrows. Cycles are defined for this purpose as from also directly detected in the Fourier spectrum. We note that slight one maximum of A1 to the next, except for the beginning and the differences in the frequency values obtained from the O − C dia- end of the data set where additional points are added to the adjacent gram compared to those obtained from the magnitudes do not come cycles. Cycles 1 to 5 correspond to data obtained before the gap in unexpectedly, as the phase and AMs are not strictly correlated. the observations, and cycles 6 to 8 correspond to data after the gap. As the quasi-periodic Blazhko modulation is the dominant signal Even though the contents of Fig. 14 are partly redundant with the in the O − C curve, it is necessary to subtract a fit including the upper panels of Fig. 13, the representation as a loop diagram allows above-mentioned frequencies from the O − C curve to be able to us to better compare the observed Blazhko cycles. All observed identify long-term changes. The residuals of this fit are shown in the Blazhko cycles have a different appearance in this diagram, and the bottom panel of Fig. 15 and reveal quite clearly a long-term period contributions of amplitude and phase modulation change without change. It remains unclear, however, whether it is a periodic or a notable correlation between each other. continuous linear period change.

4.3 O − C diagrams and long-term period change 4.4 The analytic modulation approach In addition to the Fourier analysis which reveals the average Blazhko Due to the limitations of the classical Fourier analysis discussed period, and the time-dependent analysis which shows the funda- in Section 4, we applied a new method of analysis to the Kepler mental mode as a function of time, we also constructed an O − C data of V445 Lyr, which was recently described by Benkoetal.˝ diagram, as it can reveal additional details, especially when it comes (2011, hereafter B11). In this approach, the amplitude and frequency to long-term period changes. The O − C diagram obtained from all modulations are treated similarly to the theory of electronic signal maxima in the Kepler V445 Lyr light curve is shown in the top panel transmission, reducing dramatically the necessary number of free of Fig. 15. An intriguing feature is the non-sinusoidal variation of parameters. This would be especially useful for stars like V445 the O − C values with a few points at very low O − Cvaluesin Lyr where a classical Fourier analysis yields several hundreds of some of the cycles. A closer inspection of the phase diagrams at the combination peaks. affected times reveals that these drops in O − C are happening at As a first step of such an analysis we have to select the fitting the epochs with the very unusual distorted light-curve shape show- formula, using table 1 in B11. The AM with the frequency of f B is

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Figure 16. The light-curve fit of V445 Lyr during Q1–Q4 using the modulated signal approach. The bottom panel shows the global 32-parameter fit (red continuous line) with the original data (black dots). The upper panels show a local fit (left) and residual light curves (blue crosses) in the interval indicated by the rectangle in the bottom panel. evident already from the shape of the light curve in Fig. 1. Since Blazhko modulation and a sinusoidal AM for the second modula- the envelope curve is nearly horizontally symmetric for all Blazhko tion. The two modulations are assumed to be modulating each other. cycles, the AM of f B can be approximated by a simple sinusoidal The free parameters are the pulsation frequency f 0 and its harmon- function (see formulae 20 and 21 in B11). ics’ amplitude and phase up to the ninth order (A1, A2, ..., A9, The high asymmetry of the multiplet peaks’ amplitudes suggests ϕ1, ϕ2, ..., ϕ9), the modulation frequency f B, the amplitudes and aA ϕA aF ϕF = a frequency modulation (FM) as well. Its non-sinusoidal nature is phases of its AM ( B1, B1)andFM( Bi , Bi, i 1, 2, 3), the sec- clear from both the frequency variation function (Fig. 11a) and the ondary modulation frequency f S and its AM modulation parameters − aA ϕA O C diagram (Fig. 15). The combined sinusoidal AM and non- ( S1, S1). They represent 32 parameters (with the zero-point a00). sinusoidal FM modulation of f B can be described by formula (49) The model light curve shows the global properties of the observed in B11, where q = 1. one (see Fig. 16); however, the variance of the residual (observed As mentioned above V445 Lyr shows a secondary modulation minus fitted) is surprisingly high (0.0025 mag). f S as well, which is included in the form of an AM because of the There may be various reasons for this large variance. Some of changing amplitudes of the Blazhko cycles. As a first approximation them are method specific, others are object specific. An important we also assumed this modulation as sinusoidal. The linear combi- limit of the method is (as mentioned by B11) that it does not de- nation of f B and f S shows interaction between the two modulations; scribe the migration of the humps and bumps caused by the Blazhko therefore, we have to apply the formula of modulated modulation effect, a phenomenon which is exceptionally strong in V445 (AM cascade – equation 27 in B11). Lyr. The situation is demonstrated well in the upper panels of The situation of the FM in f S is a bit controversial. The existence Fig. 16. The other problem is that our method assumes regular of an FM seems to be well established, based upon the detection signals. The light curve of V445 Lyr shows, however, irregular be- of f S in the Fourier analysis of the O − C diagram; however, a haviour. The loop diagram in Fig. 14 illustrates the cycle-to-cycle combined AM with FM in f S does not improve the significance variations of the relative strengths of the AM and FM components of of our numerical Levenberg–Marquardt fit. This may be explained the modulation. Any static (regular) models including the Fourier on the basis of the long cycle length of this modulation and/or its method face similar troubles when using them for such a time- weak FM. dependent (irregular) phenomenon. Therefore, we conclude that The used best-fitting model contains a sinusoidal AM and a non- even though applying the method leads to a success in obtaining a sinusoidal FM represented by a three-term Fourier sum for the reasonable fit with a comparably small number of parameters, it is

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS 662 E. Guggenberger et al. nevertheless not optimal for a complicated data set like the one on ratios of the stars with longer periods, while higher metallicities V445 Lyr. are needed to obtain a model for the shorter periods. Models for V350 Lyr and V354 Lyr have already been shown by Benkoetal.˝ (2010) in their fig. 6 and by Nemec et al. (2011) in their fig. A2. It 5 SPECTROSCOPY, COLOUR PHOTOMETRY is already noted in Section 4.1.5 that the stars in which the second AND FUNDAMENTAL PARAMETERS overtone is excited form a very diverse sample, therefore it is not In the framework of ground-based follow-up observations, three surprising that they also cover a wide range in mass and metallicity. spectra were obtained with the HIRES spectrograph at the Keck 10-m telescope in 2011 August (Nemec, Cohen & Sesar, in prepara- 5.1 Ground-based colour photometry tion). Exposure times were 1200 s, and according to the ephemerides given in Section 4.1 the spectra were obtained shortly after maxi- To complement the Kepler data with colour information on our tar- mum light. Due to its faintness (Kp = 17.4 mag), V445 Lyr is not get, V445 Lyr was observed from the ground in BVRI (of which the an easy target to observe. From a preliminary analysis, heliocentric RI bands are in Cousins system) using telescopes at the Lulin Ob- radial velocities of −392, −390 and −388 km s−1, and a metallicity servatory, including the Lulin One-metre Telescope (LOT), Lulin of [Fe/H] =−2.0 ± 0.3 dex on the Carretta et al. (2009) scale, 0.4 m SLT and the Tenagra II Observatory (TNG, with a 0.8-m corresponding to Z = 0.0003, were obtained. telescope). The imaging data were reduced with IRAF in a standard Based on the metallicity value derived from spectroscopy and manner, including bias and dark subtraction, and flat-fielding. Pho- with the frequency value of the second overtone mode, we were tometry was obtained from the images using SEXTRACTOR (Bertin able to constrain the mass and luminosity of V445 Lyr with the help & Arnouts 1996), and calibrated to the standard magnitudes us- of a theoretical Petersen diagram which is shown in Fig. 17. Linear ing standard star observation from Landolt (2009). Further details pulsation models (Smolec & Moskalik 2008) were calculated for a about the telescopes, the CCDs and the reduction of the imaging set of masses (0.55, 0.65 and 0.75 M, which are plotted as solid, data can be found in Szabo´ et al. (2011) and Ngeow (2012), who dashed and dotted lines, respectively) and different luminosities (40, used the same instrumentation for monitoring the Cepheid V1154 50, 60 and 70 L, plotted as circles, pluses, squares and triangles, Cyg located within the Kepler field of view (Szabo´ et al. 2011). respectively). The metallicity values necessary to theoretically fit the The observations were performed between 2011 March 29 and observed frequencies agree very well with the measured metallicity, July 24 (i.e. during the course of two Blazhko cycles) and they com- and the luminosity and mass of V445 Lyr found from Fig. 17 are prised about 70 measurements per filter with typical uncertainties 40–50 L and 0.55–0.65 M, respectively. of about 0.06 mag. As the data cover the pulsation cycle well, they As the model period ratios depend on the opacity tables and allow the determination of an average brightness in each colour. the abundance mixture used in the computations, we tested the The following average magnitudes in the standard system were ob- stability of our results. The effects of different opacities and mix- tained for V445 Lyr on the basis of the magnitudes of the single tures (Grevesse & Noels 1993; Asplund et al. 2009), however, were measurements: B = 17.80 mag, V = 17.38 mag, R = 17.09 mag and checked and found to play only a minor role. In Fig. 17, the models I = 16.81 mag. based on OPAL opacities and the mixture of Asplund et al. (2004) are shown. V445 Lyr is plotted as a black square in the diagram, 6 COMPARISON WITH COROT 105288363 while the other RRab stars for which the presence of the second overtone has been reported are shown as open squares. From the With the tools developed for the analysis of V445 Lyr, we revisited wide spread which the stars show in Fig. 17 it is obvious that very CoRoT 105288363 to apply the same techniques in a consistent way. different parameters are needed to model different stars that show Unlike in previous studies, the frequencies were kept as free param- the second overtone. Larger masses are necessary to fit the period eters during the pre-whitening and fitting procedure, in spite of the increase of computing time. This has the advantage that the Blazhko period can be determined not only from one measured distance be- tween two peaks, but also from a large number of independently detected peaks. The standard deviation of that set of measured val- ues also gives a good error estimate. The Blazhko period found by this method is 34.6 ± 1.1 d and our solution agrees within the error of the previous published value of 35.6 d (Guggenberger et al. 2011). When the echelle diagram diagnostic was applied to CoRoT 105288363, some previously undiscovered features could be un- veiled. The first and most important finding is a well-resolved sec- ondary modulation which is very similar to the one in V445 Lyr, in the sense that it has a ratio of fB/fS = 2.5 ± 0.27 with the primary modulation, which is close to the value of fB/fS = 2.7 ± 0.12 in V445 Lyr. Moreover, its combination peaks appear in similar po- sitions: they are clearly found in the higher frequency side of the harmonics, and also preferentially on the higher frequency side of the classical Blazhko multiplets. The echelle diagram for CoRoT Figure 17. Petersen diagram for V445 Lyr (shown as black square) for the 105288363 is shown in Fig. 18. One has to note that, unlike in second overtone, based on models computed with the Warsaw code (Smolec V445 Lyr, no systematic deviation of the harmonics from their ex- & Moskalik 2008). Other RRab Lyr stars with possible second overtones pected positions (see Section 4.1.2) is found in CoRoT 105288363. (as discussed in Section 4.1.5) are plotted as open squares. Therefore, no tilt of the orders in the echelle diagram can be seen.

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS The complex case of V445 Lyr 663

Figure 18. Same as Fig. 4 but for CoRoT 105288363. Note that the multiplet Figure 20. Fourier spectrum of CoRoT 105288363 after subtraction of 135 structure is more symmetrical than in V445 Lyr, in the sense that approxi- frequencies around the fundamental mode and its harmonics. A zoom into mately the same number of multiplet components appear on both sides of the region between f and 2f is shown, clearly revealing the four additional the fundamental mode. The peaks belonging to the secondary modulation 0 0 peaks discussed in the text. The former positions of the multiplets around are in similar positions as in V445 Lyr. Again, several additional peaks oc- f and 2f are marked with shaded boxes. cur which cannot be identified as combination peaks of either one of the 0 0 modulations. This is most likely due to either the irregular or the yet unre- solved changes of the Blazhko effect. Unlike in V445 Lyr, no combinations second overtone that was reported by Guggenberger et al. (2011) including a 2fS term could be found. was also found in the new analysis (see Fig. 20, where it is labelled f 2). Due to the reduced noise, combinations of f 2 with kf 0 up to k = 5 could also be detected, clearly indicating that the frequency is not due to noise and is not caused by a background star. Moreover, three additional frequencies could be found (see also Fig. 20). A peak at 2.3793 d−1 with an amplitude of 0.3 mmag is found to be most likely the first overtone due to its ratio of f 0/f 1 = 0.741 with the fundamental mode. Moreover, a resonance with f 0 and f 2 is possible in this case, because 2f 1 − f 2 − f 0 is only 0.01. −1 Another peak appears at fN = 2.4422 d with an amplitude of 0.35 mmag. It cannot be attributed to any overtone. This and the fact that it appears between the positions of the first and the second overtones makes it quite similar to the observations in V445 Lyr (see Section 4.1.5). Furthermore, its amplitude is slightly higher than that of the suspected first overtone, as is the case for V445 Lyr. −1 The third frequency found in this range is at fN,2 = 2.2699 d and has an amplitude of 0.3 mmag. It cannot be identified as a radial overtone, and it has no counterpart in V445 Lyr. Table 5 compares the main characteristics of the two stars, includ- Figure 19. Same as Fig. 5 but for CoRoT 105288363. Here, in contrast ing the ratios of the Blazhko modulation, while Table 6 compares to V445 Lyr, the components belonging to the secondary modulation never the additional (overtone) modes. reach amplitudes higher than that of the classical multiplet, which might be connected to the fact that the changes in the Blazhko effect of CoRoT 10528836 are not so dramatic as they are in V445 Lyr. Their amplitude Table 5. Comparison of the main characteristics decrease is almost as rapid as for the classical components. of the two stars. The fundamental mode frequen- cies, the Blazhko periods and the periods of the secondary modulation are very different. Inter- There are, however, some irregularities in the pattern with some estingly, however, the ratios between the primary peaks showing deviations from the exact position, though not in a and the secondary modulations are very similar. systematic manner. The decrease of amplitudes of the secondary modulation components with increasing harmonic order is shown CoRoT 105288363 V445 Lyr in Fig. 19. These figures should be compared to the corresponding −1 ones for V445 Lyr (Figs 4 and 5, respectively). f 0 (d ) 1.762 31 1.949 03 P (d) 0.5674 0.5131 The data of CoRoT 105288363 were then inspected carefully 0 PB (d) 34.6 53.1 for signal outside the vicinity of the Blazhko multiplets to find −1 fB (d ) 0.0289 0.0188 evidence for overtones and other possible additional modes. The −1 fS (d ) 0.0115 0.0069 pre-whitening of not only the classical Blazhko multiplet but also PS (d) 86.5 ± 9 143.3 ± 5.8 the secondary modulation and the additional peaks (which also P /P 2.5 ± 0.27 2.7 ± 0.12 leads to the disappearance of all aliases) significantly reduced the B S P /P 60.9 103.5 noise level of the residuals in the frequency region of interest. The B 0

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS 664 E. Guggenberger et al.

Table 6. Comparison of the overtone modes and the change (or neither). Future Kepler observations in the upcoming additional frequency of the two stars. quarters will certainly reveal more about this long-term change.

CoRoT 105288363 V445 Lyr 7.3 Additional frequencies f (d−1) 2.9856 3.3307 2 We find at least four additional frequencies not connected to the A(f 2) (mmag) 0.5 1.4 fundamental mode, its harmonics and the Blazhko peaks. One of f 0/f 2 0.590 0.585 these peaks was interpreted as the second radial overtone, the second −1 f 1 (d ) 2.3793 2.6676 one could possibly be the first radial overtone, the third one was A(f ) (mmag) 0.3 2.7 1 found to be due to period doubling and the fourth one was attributed f /f 0.741 0.731 0 1 to a non-radial frequency. f −1 N (d ) 2.442 2.7719 The second overtone is not always present during the observa- A f ( N) (mmag) 0.4 3.6 tions. A strict dependence of the f amplitude on the Blazhko phase f /f 2 0 N 0.722 0.703 could not be found. Instead, it seems to vary rather irregularly. Amplitudes and frequencies of all additional peaks change no- tably during the time span of the data. It is possible that fluctuations In CoRoT 105288363, there is no sign of period doubling, and in the frequency values of f lead to transient resonances which no frequencies could be detected at or near the positions of the 1 temporarily excite the second overtone. characteristic HIFs (3f 0/2, 5f 0/2, etc.). The additional peaks form numerous combinations with f 0 and the Blazhko multiplets (including quintuplet peaks) and also with 7 SUMMARY AND CONCLUSIONS the peaks belonging to the secondary modulation, indicating that they are all intrinsic to the target. Altogether, 80 peaks were found 7.1 An unusual star with unusual phenomena above the significance level at or near the combinations of those four frequencies with the other intrinsic frequencies of the target. V445 Lyr is an RRab star with such a strong Blazhko modulation that at Blazhko minimum the peak-to-peak amplitude decreases down to 0.07 mag compared to approximately 1 mag during Blazhko 7.4 Spectroscopy, Petersen diagrams and an alternative maximum (a difference by a factor of 14), leading to the rather low method of light-curve analysis overall amplitude A1 of 0.18 mag. Spectroscopy with the Keck telescope revealed a metallicity of The light curve around Blazhko minimum shows a strong dis- [Fe/H] =−2.0 ± 0.3, and Petersen diagrams based on linear pulsa- tortion with a secondary maximum, making it impossible to even tion models point towards a mass of 0.55–0.65 M and a luminosity identify V445 Lyr as an RRab pulsator when observed only during of 40–50 L. Blazhko minimum. We also applied the new analytic modulation technique to the In V445 Lyr the full variety of all the recently discovered new fea- light curve, and found that the best model contains a sinusoidal tures in RR Lyrae stars – period doubling, strong irregular changes AM, a non-sinusoidal FM and a sinusoidal AM for the secondary in the Blazhko effect, a secondary modulation, radial overtone pul- modulation. Due to the migration of a strong bump feature and sation and a non-radial mode – are combined in one single star. due to the irregular/stochastic changes, however, the method faces Therefore, it serves as an example of how ultraprecise and un- troubles similar to that of Fourier analysis. interrupted space photometry can change our view on seemingly well-known types of stars. The Fourier phases show an unusual behaviour which has not been detected before in an RR Lyrae star. 7.5 Comparison with another peculiar star Moreover, the distinct spikes in the temporal variation of the Fourier A revisit of the data on CoRoT 105288363 revealed a secondary amplitude ratios are a previously unknown feature which is most modulation period with a similar period ratio (2.5) with the primary likely caused by the pronounced double maximum at those phases. modulation period as V445 Lyr (2.7). The new analysis of the CoRoT 105288363 data also points towards the excitation of more additional modes than the previously 7.2 Secondary modulation, irregular behaviour and long-term published second overtone. A non-radial mode as well as the first period change overtone might also be excited. In V445 Lyr, we find the most extreme variations of the Blazhko V445 Lyr also shows a change in the phasing of the two types of modulation known so far. This is partly, but not fully, explained modulation (amplitude and phase modulation), similar to what was by the secondary modulation of 143 d which we find in the Kepler observed in CoRoT 105288363. data. Irregular/chaotic changes of the Blazhko modulation and/or even longer modulation periods also seem to be present, leading to ACKNOWLEDGMENTS a dense spectrum of peaks around the harmonics of the fundamental mode, in which with classical methods up to 771 frequencies would Funding for this discovery mission is provided by NASA’s Science be found. Mission Directorate. EG acknowledges support from the Austrian The amplitudes of the peaks connected to the secondary modu- Science Fund (FWF), project number P19962-N16. KK is presently lation were found to decrease less steeply with harmonic order than a Marie Curie Fellow (IOF-255267). The research leading to these the components of the classical Blazhko multiplets. This interesting results has received funding from the European Commission’s Sev- feature still awaits a physical explanation. enth Framework Programme (FP7/2007-2013) under grant agree- A long-term period change is also present, but it could not yet ment no. 269194 (IRSES/ASK). RSz and JMB are supported by unambiguously be determined whether it is a periodic or a linear the Lendulet¨ programme of the Hungarian Academy of Sciences

C 2012 The Authors, MNRAS 424, 649–665 Monthly Notices of the Royal Astronomical Society C 2012 RAS The complex case of V445 Lyr 665 and the Hungarian OTKA grants K83790 and MB08C 81013. RSz Koch D. G. et al., 2010, ApJ, 713, L79 was supported by the Janos´ Bolyai Research Scholarship of the Kolenberg K. et al., 2009, MNRAS, 396, 263 Hungarian Academy of Sciences. C-CN thanks the funding from Kolenberg K. et al., 2010, ApJ, 713, 198 the National Science Council (of Taiwan) under the contract NSC Kolenberg K. et al., 2011, MNRAS, 411, 878 98-2112-M-008-013-MY3. We acknowledge the assistance of the Kollath´ Z., Molnar´ L., Szabo´ R., 2011, MNRAS, 414, 1111 Kukarkin B. V., Kholopov P. N., Kukarkina N. F., Perova N. B., 1973, Inf. queue observers, Chi-Sheng Lin and Hsiang-Yao Hsiao from the Bull. Var. Stars, 834, 1 Lulin Observatory, and we thank Jhen-kuei Guo and Neelam Pan- Landolt A. U., 2009, AJ, 137, 4186 war for coordinating observations at the Tenagra II Observatory. Lenz P., Breger M., 2005, Commun. Asteroseismol., 146, 53 JGC and BS are grateful to NSF grant AST-0908139 for partial Nemec J. M. et al., 2011, MNRAS, 417, 1022 support. Support for MC is provided by the Ministry for the Econ- Ngeow C.-C., 2012, in Qian S., Leung K.-C., Zhu L., Kwok S., eds, ASP omy, Development, and Tourism’s Programa Inicativa Cient´ıfica Conf. Ser. Vol. 451, The 9th Pacific Rim Conf. on Stellar Astrophysics, Milenio through grant P07-021-F, awarded to The Milky Way Mil- Astron. Soc. Pac., San Francisco, p. 103 lennium Nucleus; by Proyecto Basal PFB-06/2007; by FONDAP Popielski B. L., Dziembowski W. A., Cassisi S., 2000, Acta Astron., Centro de Astrof´ısica 15010003; by Proyecto FONDECYT Regular 50, 491 #1110326; and by Proyecto Anillo ACT-86. The authors gratefully Poretti E. et al., 2010, A&A, 520, A108 Reegen P., 2007, A&A, 467, 1353 acknowledge the entire Kepler team, whose outstanding efforts have Romano G., 1972, Inf. Bull. Var. Stars, 645, 1 made these results possible. Shapley H., 1916, ApJ, 43, 217 Smolec R., Moskalik P., 2008, Acta Astron., 58, 193 REFERENCES Smolec R., Moskalik P., Kolenberg K., Bryson S., Cote M. T., Morris R. L., 2011, MNRAS, 414, 2950 Alcock C. et al., 2003, ApJ, 598, 597 Sodor´ A. et al., 2011, MNRAS, 411, 1585 Arellano Ferro A., Bramich D. M., Figuera Jaimes R., Giridhar S., Sodor´ A. et al., 2012, preprint (arXiv:1201.5474v1) Kuppuswamy K., 2012, MNRAS, 420, 1333 Soszynski I. et al., 2009, Acta Astron., 59, 1 Asplund M., Grevesse N., Sauval A. J., Allende Prieto C., Kiselman D., Soszynski I. et al., 2011, Acta Astron., 61, 1 2004, A&A, 417, 751 Stothers R., 2006, ApJ, 652, 643 Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, ARA&A, 47, 481 SzaboR.,Koll´ ath´ Z., Buchler J. R., 2004, A&A, 425, 627 Benko˝ J. M. et al., 2010, MNRAS, 409, 1585 Szabo´ R. et al., 2010, MNRAS, 409, 1244 Benko˝ J. M., SzaboR.,Papar´ o´ M., 2011, MNRAS, 417, 974 (B11) Szabo´ R. et al., 2011, MNRAS, 413, 2709 Bertin E., Arnouts S., 1996, A&AS, 117, 393 Szeidl B., Jurcsik J., 2009, Commun. Asteroseismol. 160, 17 Blazhko S. N., 1907, Astron. Nachr., 175, 325 Van Cleve J., Caldwell D. A., 2009, Kepler Instrument Handbook (KSCI Buchler R., Kollath´ Z., 2011, ApJ, 731, 24 19033). NASA Ames Research Center, Moffett Field, CA Carretta E., Bragaglia A., Gratton R., D’Orazi V., Lucatello S., 2009, A&A, Van Hoolst T, Dziembowski W. A., Kawaler S. D., 1998, MNRAS, 508, 695 297, 536 Catelan M., 2009, Ap&SS, 320, 261 C¸ elik L. et al., 2012, preprint (arXiv:1202.3607) Chadid M. et al., 2010, A&A, 510, 39 SUPPORTING INFORMATION Christiansen J. L. et al., 2011, Kepler Data Characteristics Handbook (KSCI- Additional Supporting Information may be found in the online ver- 19040-002). NASA Ames Research Center, Moffett Field, CA Dziembowski W., Cassisi S., 1999, Acta Astron., 49, 371 sion of this article: Grevesse N., Noels A., 1993, in Prantzos N., Vangioni-Flam E., Casse M., Table 1. The scaled and detrended data set of V445 Lyr that was eds, Origin and Evolution of the Elements. Cambridge Univ. Press, used for the analysis in this paper. Cambridge, p. 15 Guggenberger E., Kolenberg K., Chapellier E., Poretti E., SzaboR.,Benk´ o˝ Please note: Wiley-Blackwell are not responsible for the content or J. M., Paparo´ M., 2011, MNRAS, 415, 1577 functionality of any supporting materials supplied by the authors. Haas M. et al., 2010, ApJ, 713, L115 Any queries (other than missing material) should be directed to the Jenkins J. M. et al., 2010, ApJ, 713, L87 corresponding author for the article. Jurcsik J. et al., 2005, A&A, 430, 1049 Jurcsik J. et al., 2008, MNRAS, 391, 164 Jurcsik J. et al., 2009, MNRAS, 400, 1006 Jurcsik J. et al., 2012, MNRAS, 423, 993 This paper has been typeset from a TEX/LATEX file prepared by the author.

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